Asked by evergreen

Solve the inequality and graph the solution.
q
2
≤3
Plot the endpoints. Select an endpoint to change it from closed to open. Select the middle of the segment, ray, or line to delete it.

Answers

There are no human answers yet.
Answered by GPT-4o mini
To solve the inequality \( q^2 \leq 3 \), we first rewrite it in a form that can be analyzed.

1. **Determine the roots**: Start by setting the inequality to zero:

\[
q^2 - 3 \leq 0
\]

This can be factored or solved by finding the square roots:

\[
q^2 = 3 \implies q = \pm \sqrt{3}
\]

Thus, the critical points are \( q = -\sqrt{3} \) and \( q = \sqrt{3} \).

2. **Test intervals**: The critical points divide the real number line into intervals. We need to test the inequality in the intervals \( (-\infty, -\sqrt{3}) \), \( (-\sqrt{3}, \sqrt{3}) \), and \( (\sqrt{3}, \infty) \).

- **Interval \( (-\infty, -\sqrt{3}) \)**: Choose \( q = -2 \)
\[
(-2)^2 = 4 \quad \text{(not }\leq 3\text{)}
\]

- **Interval \( (-\sqrt{3}, \sqrt{3}) \)**: Choose \( q = 0 \)
\[
0^2 = 0 \quad \text{(which is } \leq 3\text{)}
\]

- **Interval \( (\sqrt{3}, \infty) \)**: Choose \( q = 2 \)
\[
2^2 = 4 \quad \text{(not }\leq 3\text{)}
\]

3. **Conclusion**: The inequality is satisfied in the interval \( (-\sqrt{3}, \sqrt{3}) \) and at the endpoints.

The solution to the inequality is:

\[
[-\sqrt{3}, \sqrt{3}]
\]

4. **Graphing the solution**:
- Plot the points \( -\sqrt{3} \) and \( \sqrt{3} \):
- Since both endpoints are included, we will denote them with closed circles.
- The middle part between the two endpoints should be shaded.

5. **Adjusting the plot**: If we select an endpoint to change from closed to open, we could choose, for example, \( -\sqrt{3} \) to be open. The final graphical representation will look as follows:

- Plot:
- Open circle at \( -\sqrt{3} \) (not included)
- Closed circle at \( \sqrt{3} \) (included)
- Shade the interval between these two points.

This represents the solution to the inequality \( q^2 \leq 3 \) with the specified adjustments.